[lmi] Dividing COI rate by one minus itself

[Greg Chicares]

Equation 3.6 in the DesRochers et al. 7702 book is:
    Exponential COI = [1-(1-q)^(1/12)]/[1-(1-(1-q)^(1/12))]
Why is the monthly rate divided by one minus itself?

The explanations in paragraph B/9 here:

  http://www.nongnu.org/lmi/7702.html
| The COI charge is assessed against all insureds who are alive
| at the beginning (not the end) of the month, and assumes that
| all deaths occur at the end (not the beginning) of the month

and on lines 218-221 here:

  http://cvs.savannah.gnu.org/viewvc/lmi/lmi/math_functors.hpp?annotate=1.16
/// Convert q to a monthly COI rate. The COI charge is assessed against
/// all insureds who are alive at the beginning of the month. Assuming
/// that deaths occur at the end of the month, the monthly-equivalent
/// q should be divided by one minus itself to obtain the COI rate.

are incorrect. The real answer has nothing to do with
 - survivorship,
 - timing of deaths,
 - immediate payment of claims, or
 - refund of unearned mortality charges.

Instead, consider:

DB  = death benefit at beginning of month
E   = expense charges deducted at beginning of month
COI = cost-of-insurance deduction
AV  = account value at beginning of month, before deduction of E or COI
i   = annual effective death benefit discount rate for NAAR calculation
q   = annual mortality rate to be converted to monthly

mv = 1 / (1 + i)^(1/12)
mq = 1 - (1 - q)^(1/12)

Deducting the COI charge at the beginning of the month increases
the amount actually at risk, suggesting an equation that has the
COI term on both sides:
    COI = [DB * mv - (AV - E - COI)] * mq
Rearranging:
    COI = [DB * mv - (AV - E)] * mq + COI * mq
    COI * (1 - mq) = [DB * mv - (AV - E)] * mq
    COI = [DB * mv - (AV - E)] * mq / (1 - mq)
suggests the definitions
    NAAR = DB * mv - (AV - E)
which is what contracts normally specify, and
    mortality-charge rate =      mq / (1 - mq)
which is the textbook's "Exponential COI".

A New York examiner once told me that this formula for converting
annual q's to monthly COI rates is commonly seen in actuarial
memoranda, and automatically accepted by NYSID.